This page lists notable examples of incomplete published mathematical proofs. Most of these were accepted as correct for several years but later discovered to contain gaps. There are both examples where a complete proof was later found and where the alleged result turned out to be false.
Results later proved rigorously
Euclid's Elements. Euclid's proofs are essentially correct, but strictly speaking sometimes contain gaps because he tacitly uses some unstated assumptions, such as the existence of intersection points. In 1899 David Hilbert gave a complete set of (second order) axioms for Euclidean geometry, called Hilbert's axioms, and between 1926 and 1959 Tarski gave some complete sets of first order axioms, called Tarski's axioms. Isoperimetric inequality. For three dimensions it states that the shape enclosing the maximum volume for its surface area is the sphere. It was formulated by Archimedes but not proved rigorously until the 19th century, by Hermann Schwarz. Infinitesimals. In the 18th century there was widespread use of infinitesimals in calculus, though these were not really well defined. Calculus was put on firm foundations in the 19th century, and Robinson put infinitesimals in a rigorous basis with the introduction of nonstandard analysis in the 20th century. Fundamental theorem of algebra (see History). Many incomplete or incorrect attempts were made at proving this theorem in the 18th century, including by d'Alembert (1746), Euler (1749), de Foncenex (1759), Lagrange (1772), Laplace (1795), Wood (1798), and Gauss (1799). The first rigorous proof was published by Argand in 1806. Dirichlet's theorem on arithmetic progressions. In 1808 Legendre published an attempt at a proof of Dirichlet's theorem, but as Dupré pointed out in 1859 one of the lemmas used by Legendre is false. Dirichlet gave a complete proof in 1837. The proofs of the Kronecker–Weber theorem by Kronecker (1853) and Weber (1886) both had gaps. The first complete proof was given by Hilbert in 1896. In 1879, Alfred Kempe published a purported proof of the four color theorem, whose validity as a proof was accepted for eleven years before it was refuted by Percy Heawood. Peter Guthrie Tait gave another incorrect proof in 1880 which was shown to be incorrect by Julius Petersen in 1891. Kempe's proof did, however, suffice to show the weaker five color theorem. The four-color theorem was eventually proved by Kenneth Appel and Wolfgang Haken in 1976. Schröder–Bernstein theorem. In 1896 Schröder published a proof sketch which, however, was shown to be faulty by Alwin Reinhold Korselt in 1911 (confirmed by Schröder). Jordan curve theorem. There has been some controversy about whether Jordan's original proof of this in 1887 contains gaps. Oswald Veblen in 1905 claimed that Jordan's proof is incomplete, but in 2007 Hales said that the gaps are minor and that Jordan's proof is essentially complete. In 1905 Lebesgue tried to prove the (correct) result that a function implicitly defined by a Baire function is Baire, but his proof incorrectly assumed that the projection of a Borel set is Borel. Suslin pointed out the error and was inspired by it to define analytic sets as continuous images of Borel sets. Dehn's lemma. Dehn published an attempted proof in 1910, but Kneser found a gap in 1929. It was finally proved in 1956 by Christos Papakyriakopoulos. Hilbert's sixteenth problem about the finiteness of the number of limit cycles of a plane polynomial vector field. Henri Dulac published a partial solution to this problem in 1923, but in about 1980 Écalle and Ilyashenko independently found a serious gap, and fixed it in about 1991. In 1929 Lazar Lyusternik and Lev Schnirelmann published a proof of the theorem of the three geodesics, which was later found to be flawed. The proof was completed by Werner Ballmann about 50 years later. Littlewood–Richardson rule. Robinson published an incomplete proof in 1938, though the gaps were not noticed for many years. The first complete proofs were given by Marcel-Paul Schützenberger in 1977 and Thomas in 1974. Class numbers of imaginary quadratic fields. In 1952 Heegner published a solution to this problem. His paper was not accepted as a complete proof as it contained a gap, and the first complete proofs were given in about 1967 by Baker and Stark. In 1969 Stark showed how to fill the gap in Heegner's paper. In 1954 Igor Shafarevich published a proof that every finite solvable group is a Galois group over the rationals. However Schmidt pointed out a gap in the argument at the prime 2, which Shafarevich fixed in 1989. Nielsen realization problem.